22-Mec-B2 Environmental Control in Buildings · December 2016

Question 3 of 8: Annual energy and running cost by the degree-day method

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. Professional Engineers of Ontario / Engineers Canada annual examination 07-Mec-B2 Environmental Control in Buildings, December 2016, three hours, open book. Eight problems of 20 points each; candidates are required to solve five, and all questions carry the same value. Psychrometric charts and an R-22 p-h diagram are appended to the paper. All eight problems are solved here.

Reference texts for this subject.

W. P. Jones, Air Conditioning Engineering, 5th ed., Butterworth-Heinemann — the standard reference for this examination code; Ch. 2–3 (psychrometry), Ch. 5–6 (heating and cooling loads), Ch. 15 (fans and duct design).
McQuiston, Parker & Spitler, Heating, Ventilating and Air Conditioning: Analysis and Design, 6th ed., Wiley — Ch. 3 (moist air), Ch. 5 (heat transmission in building structures), Ch. 8 (energy estimating and the degree-day method), Ch. 12–13 (fluid flow, fans and duct design).
ASHRAE Handbook — Fundamentals (2021) — Ch. 1 (psychrometrics), Ch. 14 (climatic design information), Ch. 18 (non-residential cooling and heating load calculations), Ch. 21 (duct design), Ch. 26 (heat, air and moisture control), Ch. 30 (thermophysical properties of refrigerants).
Moran, Shapiro, Boettner & Bailey, Fundamentals of Engineering Thermodynamics, 9th ed., Wiley — Ch. 10 (vapour-compression refrigeration and heat pumps).
ANSI/ASHRAE Standard 62.1, Ventilation for Acceptable Indoor Air Quality; ANSI/ASHRAE Standard 55, Thermal Environmental Conditions for Human Occupancy.
Canadian frame: National Building Code of Canada 2020 and its Appendix C design temperatures; National Energy Code of Canada for Buildings 2020; Environment and Climate Change Canada degree-day normals for Toronto and Ottawa; CSA B52 Mechanical Refrigeration Code; Canada Green Building Council LEED Canada and the CaGBC Zero Carbon Building Standard for Problem 5.

Check: assumptions carried through this paper. Cover-page instruction 1 invites a clear statement of any assumption. Standard barometric pressure of 101.325 kPa is used throughout; moist-air properties follow the ASHRAE Handbook — Fundamentals Ch. 1 formulation (Hyland–Wexler saturation pressure, so results agree with the appended chart to chart-reading accuracy rather than being read off it); R-22 properties are on the IIR datum and agree with the appended p-h diagram. Problem-specific assumptions — the coil bypass factor, climate and degree-day data, fuel prices and equipment efficiencies, duct roughness, and the CLTD / SCL / CLF table entries — are stated where they are first used.

Question 3: Annual energy and running cost by the degree-day method (20 points)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A 5,000 ft² downtown Toronto office with the design loads and design conditions below, plus a steady winter internal gain.

Given data, Problem 3
Quantity	Symbol	Value
Floor area	A	50 ft × 100 ft = 5,000 ft²
Design heating load	q_h	280,000 Btu/h at 75 / −5 °F
Design cooling load	q_c	125,000 Btu/h at 78 / 91 °F
Average winter internal gain	q_int	8 kW = 27,297 Btu/h
Natural gas price	—	$4.00 per thousand ft³
Electricity price	—	$0.10 per kWh
Toronto heating degree-days, base 65 °F	HDD₆₅	7,002 °F·day (assumed)
Toronto cooling degree-days, base 65 °F	CDD₆₅	707 °F·day (assumed)

Find. The annual heating and cooling energy requirements by the degree-day method, the annual energy cost of each of the two proposed systems, and recommendations for reducing them.

Check: climate and equipment assumptions. The paper supplies no degree-day data, so Toronto Pearson normals are used: HDD₆₅ = 7,002 and CDD₆₅ = 707 °F·day (Environment and Climate Change Canada / ASHRAE Fundamentals Ch. 14, converted from 3,890 and 393 °C·day at base 18.3 °C). The variable-base sensitivity is taken as 245 °F·day per °F of base near 65 °F, which is simply the number of days a year colder than that base. Equipment: electric resistance at COP 1.0, gas furnace at 80 % seasonal efficiency, natural gas at 1,000 Btu/ft³ higher heating value (so one MCF equals one MMBtu), and a "high efficiency" air conditioner at SEER 16.

Approach. Convert each design load into a building conductance, use the internal gain to move the heating balance point away from the nominal 65 °F base, integrate the demand with degree-days to that balance point, and then divide by each system's seasonal efficiency and multiply by its fuel price.

Turn the design loads into conductances. The degree-day method needs a heat-loss rate per degree, not a peak load:$$(UA)_{h}=\frac{q_{h}}{t_{i}-t_{o}}=\frac{280{,}000}{75-(-5)}=3,500\ \text{Btu/h}\cdot{}^\circ\text{F}^{-1}$$$$(UA)_{c}=\frac{q_{c}}{t_{o}-t_{i}}=\frac{125{,}000}{91-78}=9,615\ \text{Btu/h}\cdot{}^\circ\text{F}^{-1}$$The summer figure is much the larger because the design cooling load is mostly solar and internal gain rather than conduction — a warning that the cooling estimate will be the weaker of the two.
Find the heating balance point. The office does not need heat until the loss exceeds the internal gain:$$\Delta t_{\text{bal}}=\frac{q_{\text{int}}}{(UA)_{h}}=\frac{27,297}{3,500}=7.80\ \text{F}^\circ,\qquad t_{\text{bal}}=75-7.80=67.20\ ^\circ\text{F}$$This is the step the 8 kW datum exists for. Note that the balance point lands above the conventional 65 °F base, because the office is held at 75 °F rather than the 70 °F that the base-65 convention assumes; using HDD₆₅ unadjusted would therefore understate the heating requirement.
Correct the degree-days to the balance point. At 245 °F·day per °F of base,$$\mathrm{HDD}_{t_{\text{bal}}}=7{,}002+245\times(67.20-65)=7,541\ ^\circ\text{F}\cdot\text{day}$$which corresponds to $\mathrm{EFLH}=24\,\mathrm{HDD}/(t_{i}-t_{o})=2,262$ equivalent full-load hours — a thoroughly plausible figure for southern Ontario and a useful sanity check on the whole calculation.
Annual heating energy delivered.$$Q_{h}=24\,(UA)_{h}\,\mathrm{HDD}_{t_{\text{bal}}}=24\times3,500\times7,541$$$$\boxed{\ Q_{h}=633.5\ \text{MMBtu/yr}\ }$$The internal gain is worth 160.5 MMBtu/yr of that: without it the balance point would be the room temperature itself and the demand would rise to $24\times3,500\times9,452/10^{6}$ MMBtu. At 126.7 kBtu/ft²·yr for heating alone this is a very leaky building by modern standards, which is the first clue for the recommendations.
Annual cooling energy removed. Applying the same method on the summer side, with the nominal base of 65 °F:$$Q_{c}=24\,(UA)_{c}\,\mathrm{CDD}_{65}=24\times9,615\times707$$$$\boxed{\ Q_{c}=163.2\ \text{MMBtu/yr}\ }$$This is the least trustworthy number in the problem: it implies 1,305 equivalent full-load cooling hours, roughly twice what a southern-Ontario office actually runs, because the small 13 F° design difference makes $(UA)_{c}$ absorb solar and internal gains that do not in fact scale with outdoor temperature. It is reported as the degree-day answer the question asks for, with the caveat attached.
Cost the two systems. Electric resistance delivers one kilowatt-hour of heat per kilowatt-hour bought, whereas the gas furnace burns 633.5/0.80 MMBtu of gas:$$\text{baseboard: }\frac{633.5\times10^{6}}{3412}\times\text{\$}0.10=\text{\$}18,565\ \text{per year}$$$$\text{furnace: }\frac{633.5}{0.80}\times\text{\$}4.00=791.8\ \text{MCF}\times\text{\$}4.00=\text{\$}3,167\ \text{per year}$$The air conditioner is common to both: $163.2\times10^{6}/16=10,197$ kWh, or $1,020 a year. Unit heat costs make the gap plain — $29.31 per MMBtu delivered by resistance against $5.00 per MMBtu delivered by an 80 % furnace.
Total annual energy cost.$$\boxed{\ \text{(a) baseboard + AC}=\text{\$}19,585\ \text{per year}\ }$$$$\boxed{\ \text{(b) gas furnace + AC}=\text{\$}4,187\ \text{per year}\ }$$Option (b) saves about $15,398 a year, and the heating bill alone falls by a factor of 5.86.

The balance-point construction on the left and the annual running costs on the right. The internal gain shifts the point at which heating is needed down from the room temperature to 67.20 °F; the cost bars show how completely the heating fuel choice dominates the annual bill.

Recommendations. The gas furnace is the obvious choice on running cost, but the more useful advice concerns the building rather than the fuel. A conductance of 3,500 Btu/h·°F over 5,000 ft² is 56 Btu/h·ft² at design, two to three times what NECB 2020 or the Toronto Green Standard would allow for new construction, so envelope work — air sealing first, then glazing and insulation — attacks the largest term directly and cuts both bills at once. A condensing furnace at 95 % AFUE would bring the gas cost down to about $2,667 a year for a modest premium. A heat-recovery ventilator recovers 60–80 % of the ventilation heat, which in Toronto is a large share of the seasonal total. Night and weekend setback exploits the fact that an office is unoccupied about two-thirds of the hours in the heating season. On the cooling side, exterior or interpane shading and daylight-linked lighting control cut the gain before it has to be removed, and an economiser gives free cooling through the long Toronto shoulder seasons. Finally, a cold-climate air-source heat pump in a dual-fuel arrangement with the gas furnace is now worth costing: at Ontario electricity prices it competes with gas above roughly −10 °C and, on Ontario's largely non-emitting grid, it reduces emissions substantially even where it does not reduce the bill.

Problem 3 — results
Quantity	Value
Heating conductance (UA)_h	3,500 Btu/h·°F
Cooling conductance (UA)_c	9,615 Btu/h·°F
Heating balance point	67.20 °F
Degree-days to the balance point	7,541 °F·day (2,262 EFLH)
Annual heating energy delivered	633.5 MMBtu/yr = 185,649 kWh
Annual cooling energy removed	163.2 MMBtu/yr (10,197 kWh at SEER 16)
(a) Electric baseboard + AC	$18,565 + $1,020 = $19,585 per year
(b) Gas furnace + AC	$3,167 + $1,020 = $4,187 per year
Annual saving of (b) over (a)	$15,398

Topic. Seasonal energy estimation by the balance-point degree-day method.

Key relations. $UA=q_{\text{design}}/\Delta t_{\text{design}}$; balance point $t_{\text{bal}}=t_{i}-q_{\text{int}}/UA$; seasonal demand $Q=24\,UA\,\mathrm{DD}_{t_{\text{bal}}}$; equivalent full-load hours $\mathrm{EFLH}=24\,\mathrm{DD}/\Delta t_{\text{design}}$; fuel $=Q/\eta$.

Why this works. Degree-days are simply the integral of $(t_{\text{base}}-t_{o})$ over the hours when the outdoor temperature is below the base, so multiplying by a constant $UA$ and by 24 turns a table of climate normals into an annual energy. The method rests on the load being linear in outdoor temperature and on the balance point being the right base; the conventional 65 °F base is a stand-in for a 70 °F room with 5 F° of internal gain, and whenever the real building differs — as it does here, with a 75 °F room and 8 kW of gain — the base must be moved to the computed balance point. Expressing the answer as equivalent full-load hours is the cheapest sanity check available: 2,000–2,500 hours is right for southern Ontario heating, and the 1,305 cooling hours this method returns is visibly not.

Common pitfall. Using HDD₆₅ unadjusted and then subtracting the internal gain again, which double-counts the gain; and trusting the cooling result, where the degree-day method is genuinely poor because solar and internal gains do not track outdoor temperature. For cooling, bin or hourly simulation methods, or published equivalent-full-load hours, are the professional answer.

Source. McQuiston, Parker & Spitler, HVAC: Analysis and Design, 6th ed., Ch. 8 (energy calculations); ASHRAE Handbook — Fundamentals (2021), Ch. 14 (climatic design information) and Ch. 19 (energy estimating and modelling methods).

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